Method and apparatus for measuring and orienting golf club shaft

ABSTRACT

The preferred orientation, or planar oscillation plane, of a golf club shaft is located by measuring the oscillation of the shaft when an impulse is applied. Preferably, the out-of-plane oscillation is measured at a large number of angular positions about the shaft axis, and the principal planar oscillation plane is identified by that pair of opposed angular positions in which the out-of-plane oscillation is smallest. The location of the preferred orientation may be marked on the shaft and used to assemble a golf club with the planar oscillation plane in a predetermined orientation. The straightness of the shaft can also be determined by deriving its spring constant from its oscillation frequency and then measuring the restoring force when the shaft is deflected by the same nominal amount at different angular positions; differences in restoring force can be attributed to differences in actual deflection distance resulting from lack of straightness.

CROSS REFERENCE TO RELATED APPLICATIONS

This is a division of commonly-assigned U.S. patent application Ser. No.10/037,701, filed Nov. 9, 2001, now U.S. Pat. No. 6,915,695, whichclaims the benefit of U.S. Provisional Patent Applications Nos.60/247,141 and 60/263,489, filed Nov. 10, 2000 and Jan. 22, 2001,respectively.

BACKGROUND OF THE INVENTION

This invention relates to measuring and orienting a golf club shaft.More particularly, this invention relates to a method and apparatus forautomatically and reliably identifying the location of a planaroscillation plane, and particularly the principal planar oscillationplane, of a golf club shaft and for aligning that planar oscillationplane in a desired orientation, as well as for determining parameters ofthe golf club shaft, such as roundness, stiffness and straightness, thatcharacterize golf club performance.

When a golfer swings a golf club, the shaft of the golf club bends ortwists, especially during the downswing. The direction the shaft bendsor twists is dependent on how the golfer loads or accelerates the club,but the bending or twisting direction and magnitude also are dependenton the stiffness of the shaft. If a shaft is soft, it will bend or twistmore during a given downswing than if it is stiff. Additionally if ashaft exhibits different transverse stiffness in different planes—i.e.,the stiffness, roundness and straightness of the shaft are notsymmetric—the shaft will bend or twist differently depending upon inwhich plane (direction) it is loaded.

Immediately prior to the impact of the head of a golf club with a golfball, the shaft of the golf club goes through significant vibratorymovements in both the toe up/down direction (plane perpendicular to thehit direction) and in the lead/lag direction (plane parallel to the hitdirection). Research has shown the shaft of a golf club vibrates up anddown in the toe up/down direction immediately prior to impact with thegolf ball. This up and down movement, known as “vertical deviationoscillation,” “vertical deflection oscillation” or “droop oscillation,”can be as large as ±1.5 inch (±3.8 cm). Inconsistent bending or twistingmakes it more difficult for the golfer to reproduce the downswing shaftbending or twisting from club to club, thereby resulting in lessconsistent impact repeatability within the set. Because any inconsistentbending or twisting due to asymmetric shaft behavior immediately priorto impact is substantially impossible for the golfer to correct with hisor her swing, any reduction in the aforementioned oscillationimmediately prior to impact will help the golfer improve his or herimpact repeatability, thereby enhancing performance. This is true forgolfers of all skill levels.

In addition, a golf club, immediately prior to impact, “springs” forwardin the direction of the shot. This is commonly referred to as the “kick”of the shaft. If it is possible to analyze and orient a shaft in a waythat the kick direction of vibration is stable, this shaft positionwould improve the golfer's ability to repeat the impact position withthe ball. In other words the shaft would have less of a tendency to“bob” up and down immediately prior to impact thereby improving impactrepeatability.

Inconsistent bending or twisting contributes to movements of the clubhead that would not be present if the shaft had been perfectlysymmetric. Golf club shaft manufacturers attempt to build shafts withsymmetric stiffness to minimize inconsistent bending or twisting duringthe swing, but as a result of manufacturing limitations it is difficultto build a perfectly symmetric golf club shaft. Specifically, it is wellknown that, as a result of irregularities or variations in materials ormanufacturing processes, golf club shafts have a preferred angularorientation. For example, it is sometimes said that a golf club shafthas a “spine” whose orientation may be significant. (See, e.g., U.S.Pat. Nos. 4,958,834 and 5,040,279, which are hereby incorporated byreference in their entireties.) Therefore, substantially all golf clubshafts exhibit some degree of asymmetry which results in some degree ofinconsistent bending or twisting during the swing.

The asymmetry of golf club shafts can result from nonsymmetrical crosssections (shafts whose cross sections are not round or whose wallthicknesses are not uniform), shafts that are not straight, or shaftswhose material properties vary around the circumference of the shaftcross section. Because it is substantially impossible to build aperfectly symmetric golf club shaft and the objective is to minimizeinconsistencies from club to club in a golf club set and from set to setwithin a brand, it makes sense, if possible, to analyze each golf clubshaft in a set of golf clubs to understand its asymmetric bending ortwisting behavior and construct the golf clubs in the set to maximizeconsistency from club to club within a set and from set to set within abrand.

It has been recognized—e.g., in above-incorporated U.S. Pat. No.5,040,279—that although substantially all golf club shafts exhibit somedegree of asymmetry, substantially every golf club shaft exhibits atleast one orientation in which, when the shaft is clamped at itsproximal, or handle, end and displaced at the tip, the resultantvibration of the shaft will remain substantially planar. That is, theshaft will remain substantially in a single plane and the tip of theshaft will vibrate back and forth substantially along a line.

It is also recognized in above-incorporated U.S. Pat. No. 4,958,834 thatthe construction of all golf clubs within a set with their respectiveplanar oscillation planes (“POPs”) oriented in the same angulardirection relative to their respective club faces will exhibit lessinconsistency in shaft bending or twisting during the downswing than aset that has been haphazardly or randomly constructed. In particular, aset of golf clubs normally will function best if the respectivepreferred angular orientations of the respective golf club shafts arealigned in the “hit direction”—i.e., substantially perpendicularly tothe respective golf club faces.

However, heretofore there has not been any convenient automated way todetermine with consistency the parameters of a golf club shaft thatwould allow manufacturers or others to predict the performance of a golfclub shaft. And while copending, commonly-assigned U.S. patentapplication Ser. No. 09/494,525, filed Feb. 1, 2000, which is herebyincorporated by reference in its entirety, showed a method and apparatusfor determining the preferred angular orientation of a golf club shaft,that method and apparatus were partly manual, and relied on an iterativetechnique which, in identifying a planar oscillation plane, couldidentify a planar oscillation plane other than the principal planaroscillation plane. It would be desirable to be able to provide a methodand apparatus for quickly and reliably determining the preferred angularorientation of a golf club shaft. It also would be desirable to be ableto provide a method and apparatus for using the determination of thepreferred angular orientation to automatically assemble golf clubs witheach respective golf club shaft consistently aligned relative to therespective club face. It would further be desirable to be able todetermine parameters of a golf club shaft to allow prediction of golfclub performance.

SUMMARY OF THE INVENTION

It is an object of this invention to attempt to provide a method andapparatus for quickly and reliably determining the preferred angularorientation of a golf club shaft.

It is also an object of this invention to attempt to provide a methodand apparatus for using the determination of the preferred angularorientation—e.g., the planar oscillation plane, esp. the principalplanar oscillation plane—to assemble golf clubs (manually orautomatically) with each respective golf club shaft consistently alignedrelative to the respective club face.

It is further an object of this invention to attempt to determineparameters of a golf club shaft to allow prediction of golf clubperformance.

In accordance with the present invention, there is provided a method ofdetermining a preferred angular orientation of a golf club shaft about alongitudinal axis thereof, where the golf club shaft has a proximal endfor gripping by a golfer and a distal end for attachment to a golf clubhead. According to the method, the proximal end of said golf club shaftis immobilized, and vibratory motion of the distal end of the golf clubshaft is initiated in a direction other than parallel to thelongitudinal axis. The vibratory motion is analyzed, and from theanalyzed vibratory motion the preferred angular orientation iscalculated. The golf club shaft can then be marked to indicate thepreferred angular orientation. In a further method according to theinvention, the mark on the shaft indicating the preferred angularorientation can be used to manually or automatically assemble a golfclub with the golf club shaft in a predetermined alignment relative tothe face of the golf club head.

Apparatus for determining the preferred angular orientation, and forassembling golf clubs, are also provided.

In a particularly preferred method and apparatus, the vibratory motionof the golf club shaft is analyzed at a plurality of angular positionsabout the longitudinal axis of the shaft. The greater the number ofpositions, the more accurately the planar oscillation plane—andparticularly the principal planar oscillation plane—can be detected. Inaddition, at each position, the vibration frequency of the shaft, whichis a measure of its stiffness, can be determined. In addition, if theshaft is deflected from its longitudinal axis, then by measuring, ateach angular position, the restoring force opposing the deflection, andthe amount of shaft deflection, one can determine the straightness ofthe shaft or, more particularly, the degree to which it is not straight.Roundness, straightness and stiffness are parameters that characterizethe performance of a golf club shaft, and shaft manufacturers havesought ways to accurately determine these parameters.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other objects and advantages of the invention will beapparent upon consideration of the following detailed description, takenin conjunction with the accompanying drawings, in which like referencecharacters refer to like parts throughout, and in which:

FIG. 1 is a diagram in which a flexible golf shaft is modeled as a massto which springs are attached;

FIG. 2 shows the horizontal and vertical displacement, seen end-on, ofthe shaft of FIG. 1 as a function of time, over two oscillation cyclesafter an impulse is delivered to cause the shaft to oscillate;

FIG. 3 shows the motion diagramed in FIG. 2 as a phase plot;

FIG. 4 shows the motion of the shaft as a phase plot, after fourteenoscillation cycles;

FIG. 5 shows the motion diagramed in FIG. 4, but as a function of time;

FIG. 6 is a perspective view of a first preferred embodiment ofapparatus according to the present invention for determining thepreferred orientation of a golf club shaft;

FIG. 7 is a perspective view of a shaft testing assembly of theapparatus of FIG. 6;

FIG. 8 is a perspective view of a shaft holding and rotating assembly ofthe apparatus of FIGS. 6 and 7;

FIG. 9 is a perspective view of a measurement assembly of the apparatusof FIGS. 6–8;

FIG. 10 is a perspective view of a tip mass and sensor assembly of theapparatus of FIGS. 6–9;

FIG. 11 is a view similar to FIG. 7 with a golf club shaft mounted inthe apparatus;

FIG. 12 is an end elevational view, taken from line 12—12 of FIG. 11,but with the golf club shaft deflected in preparation for oscillationaccording to the invention;

FIG. 13 is perspective view of the apparatus of FIGS. 6–10 with amarking assembly included;

FIG. 14 is a flow diagram of a preferred embodiment of a methodaccording to the invention for location the preferred orientation of agolf club shaft;

FIG. 15 is a flow diagram of a load test performed according to theinvention as part of the method of FIG. 14;

FIG. 16 is a flow diagram of a “logo up” comparison test performedaccording to the invention as part of the method of FIG. 14;

FIG. 17 is a flow diagram of a planar oscillation plane locating testperformed according to the invention as part of the method of FIG. 14;

FIG. 18 is a perspective view of a second preferred embodiment ofapparatus according to the present invention for determining thepreferred orientation of a golf club shaft;

FIG. 19 is a side elevational view of a measurement assembly of theapparatus of FIG. 18;

FIG. 20 is an end elevational view of the measurement assembly of FIG.19;

FIG. 21 is a plot of maximum out-of-plane acceleration or displacement,as a function of rotational angle, of a shaft tip during measurementusing the apparatus of FIG. 18;

FIG. 22 is a plot, as a function of angle, of the deviation of the tipcenter of a typical golf club shaft from a longitudinal axis passingthrough the center of the shaft butt, as measured using the apparatus ofFIG. 18;

FIG. 23 is a diagrammatic view of apparatus according to the inventionfor assembling golf clubs;

FIG. 24 is a close-up view of an assembly station of the apparatus ofFIG. 23;

FIG. 25 is a sample of a printout used to demonstrate the results oftesting of an individual shaft;

FIG. 26 is a perspective view of an alternative preferred embodiment ofa tip mass assembly according to the invention;

FIG. 27 is a schematic elevational view of the tip mass assembly of FIG.26 mounted on the tip of a golf club shaft in the rest position inapparatus according to the invention; and

FIG. 28 is a schematic elevational view, similar to FIG. 27, of the tipmass assembly of FIG. 26 mounted on the tip of a golf club shaft in adisplaced position in apparatus according to the invention.

DETAILED DESCRIPTION OF THE INVENTION

If a golf club shaft is immobilized at its handle end and displaced in adirection perpendicular to its longitudinal axis, then if thedisplacement direction lies in a planar oscillation plane of the shaft,the shaft will vibrate in that plane and, viewed end on, the distal tipof the shaft will oscillate back and forth along a line. Forconvenience, that line can be referred to as the x-axis. However, if thedisplacement direction is in a plane other than a planar oscillationplane, the distal tip of the shaft will vibrate in a motion havingcomponents along the x-axis as well as along an axis perpendicular tothe x-axis, which for convenience can be referred to as the y-axis. Thismotion could be described as an “orbital” motion, although rather thantracing a single ellipse or other closed curve, the tip will move withinan envelope such that, if the motion would not damp out (as it inreality does), it would be expected that the tip eventually would movethrough every point within that envelope.

As described below, by observing the tip vibration of the shaft, one cancalculate mathematically the orientation of the planar oscillation planeor planes. Having located the planar oscillation plane or planes, onecan then assemble a golf club, orienting the shaft relative to the golfclub head so that a planar oscillation plane, and particularly theprincipal planar oscillation plane, is lined up along the “hitdirection”—i.e., substantially perpendicular to the hitting face of theclub head—or 180° opposite to that direction. It is also possible,having located a planar oscillation plane of a golf club shaft, to alignthat planar oscillation plane relative to the golf club head not alongthe hit direction, but in another predetermined direction.

For example, it may be desirable to align the shaft for a particulargolfer to correct or induce a hook or a slice. Thus, for a right-handedgolf club, to induce a hook or correct a slice, one would rotate theshaft counterclockwise (looking down the shaft toward the club head),and to induce a slice or correct a hook, one would rotate the shaftclockwise. For a left-handed golf club, the directions of rotation wouldbe reversed. The amount of rotation preferably should be less than about90°.

It has been observed empirically that a golf club shaft functions asthough it is “harder” in one direction along any planar oscillationplane than it is in the opposite direction along that planar oscillationplane. This “harder” side of the planar oscillation plane of the shaftcan be referred to as the “hard” or “forward” side of the planaroscillation plane, while the less hard side, 180° opposite the hardside, can be referred to as the “soft” or “rear” side of the planaroscillation plane. It has also been observed that orienting a planaroscillation plane perpendicular to the club head face may yielddifferent results as compared to a haphazard or random alignment. It hasfurther been observed that aligning the planar oscillation planeperpendicular to the club head face with the hard side of the planaroscillation plane facing toward the club head face yields a differentresult than aligning the planar oscillation plane perpendicular to theclub head face with the soft side of the planar oscillation plane facingtoward the club head face. Moreover, if every golf club in a set of golfclubs is similarly aligned, there is a greater likelihood that the userof those clubs will be able to achieve more uniform and consistentresults across all golf clubs in the set, which can be expected toresult in performance enhancement.

In addition, it has been observed empirically that a golf club shaft mayhave several planar oscillation planes. However, it has been found thatthere is a principal planar oscillation plane (“PPOP”), which also maybe referred to as the plane of uniform repeatability (“PURE”). Golfclubs aligned based on the principal planar oscillation plane can beexpected to result in optimal performance enhancement.

Although it is possible to derive the orientation of the planaroscillation plane or planes precisely using mathematical techniquesbased on data collected by displacing the shaft tip and allowing theshaft to vibrate, it is computationally simpler to derive theorientation by an iterative technique as described below. The iterativetechnique can be carried out using equipment that induces vibration of agolf club shaft at a plurality of angular orientations, measuring thetip oscillation at each orientation. The equipment can be partiallymanually operated, in that the shaft is rotated manually to a neworientation for measurement, or equipment can be used in which therotation of the shaft to each subsequent position, after completion ofmeasurements at the prior position, is performed automatically. If therotation of the shaft is performed automatically, the equipment can beoperated more quickly, allowing the golf club shaft to be measured inmore angular orientations, which can be expected to produce a moreaccurate determination of the principal planar oscillation plane.

The preferred direction of a planar oscillation plane—i.e., in the caseof the principal planar oscillation plane, the “hard” side of the golfclub shaft—cannot be determined mathematically from mere observation ofthe shaft tip. Therefore, in a preferred embodiment of the invention,the handle or butt end of the golf club shaft is immobilized, the tip ofthe shaft is displaced perpendicular to the longitudinal axis, and therestoring force—i.e., the force tending to move the tip back to itsneutral position—is measured while the shaft is rotated, from the handleend, through at least about 360°. The angle at which the restoring forceis greatest is an indication of the hard side of the shaft. Althoughthis angle usually will not align precisely with the orientation of theprincipal planar oscillation plane, it will indicate which of the twopossible orientations of the principal planar oscillation planecorresponds to the hard side of the principal planar oscillation plane.Moreover, starting one's analysis at the angle of maximum load can beexpected to lead one to find the principal planar oscillation planerather than one of the other planar oscillation planes of the shaft.This is particularly true in an embodiment in which the shaftoscillation is measured at only a relatively few angular positions, suchas the partially manual embodiment discussed above. The initial positionis less important in an embodiment in which measurements are taken atrelatively more angular positions, such as the embodiment describedabove in which rotation of the golf club shaft from position to positionis carried out automatically. In either case, the starting orientationalso can be selected arbitrarily.

Once the preferred angular orientation of the golf club shaft has beendetermined, one or more marks preferably are made on the shaft toindicate the preferred angular orientation. The mark or marks may bemade at the location of a planar oscillation plane, or at apredetermined relative position with respect to a planar oscillationplane. Each mark can be made using ink or paint, or can be etched intothe surface of the shaft using another technique, such as a mechanical,electrostatic or laser marking technique, or a marked label (e.g., asticker or decal) can be applied. Once the mark or marks have been made,they can be used to align the shaft relative to a golf club head whenassembling a golf club, so that the marked planar oscillation plane ofthe golf club shaft is substantially perpendicular to, or at some otherdesired orientation with respect to, the club head face.

The alignment of the shaft to the club head can be performed manually.Preferably, alignment is facilitated by providing a marking on the clubhead as well, on or near the hosel or bore, to which a marking on theshaft can be aligned to form a properly “spine-aligned” golf club.Alternatively, in another preferred embodiment, an assembly machinemates a golf club head to a golf club shaft, matching up the alignmentmarkings in the process. In this embodiment, the golf club head can beattached to the shaft immediately after determination of the preferredangular orientation of the shaft, with the shaft still in the chuck ofthe planar oscillation plane locating station (in that case, theapplication of a visible mark to the shaft exterior can be omitted,although it would still be useful for later repair operations when theclub is disassembled). Alternatively, in a second variant of thisembodiment, the shaft can be removed from the planar oscillation planelocating station and moved to a club assembly station. This variantbetter accounts for any speed differential between the planaroscillation plane locating process and the club assembly process. If theplanar oscillation plane locating process is faster than the clubassembly process, more club assembly stations than planar oscillationplane locating stations can be provided. If the club assembly process isfaster than the planar oscillation plane locating process, more planaroscillation plane locating stations than club assembly stations can beprovided. In either case, it is preferable to provide a hopper or otherintermediate station for holding spine-aligned shafts between the planaroscillation plane locating station and the club assembly station.Normally, one would expect few shafts to be held in the hopper, but iffor some reason there is a breakdown or other bottleneck at ordownstream of the club assembly station or stations, the hopper canserve, until it is full, as a reservoir to accept shafts from the planaroscillation plane locating station or stations.

In addition to locating the planar oscillation plane(s) of golf clubshafts, whether for re-alignment of existing golf clubs or for theassembly of new golf clubs, the present invention, and particularly theembodiment described above in which rotation of the shaft from oneangular position to another for measurement purposes is automated,allowing measurements to be taken at more angular positions, providesthe ability to measure certain characteristics of a shaft that can beused to monitor the shaft manufacturing process and the quality of theresulting shafts. These measurements can provide a qualificationstandard for golf shaft manufacture.

Specifically, at each angular position, when the shaft if deflected andallowed to vibrate, the frequency of the shaft vibration can bemeasured. This can be done simply by counting the number times in agiven time interval that the vibrating shaft passes a fixed point. Oneway to perform that counting function is to provide a light source and aphotodetector and to count the number of times during the given timeinterval that the beam from the light source is interrupted by thevibrating shaft. In an alternative preferred method, the oscillationsrecorded by accelerometer data (see below) within a particular timeinterval can be counted.

Once the characteristic vibration frequency has been determined, thespring constant of the shaft, which is a measure of its stiffness, canbe approximated by treating the shaft as a prismatic beam of mass M andderiving the spring constant, k, from the frequency, f, using therelationship f≈(k/M)^(0.5). The stiffness of the shaft can then becharacterized by the value of k at each angle, all as described in moredetail below.

At each angular position, a load test can also be administered, bydeflecting the shaft through a fixed distance, d, transverse to itslongitudinal axis and measuring the restoring force, F, generated. Fromthe force, F, and the spring constant, k, determined above, one candetermine the deviation, δ, which is a measure of the straightness ofthe shaft, from the relation F/k=d+δ. The straightness of the shaft canthen be characterized by the value of δ at each angle, all as describedin more detail below.

The invention will now be described with reference to FIGS. 1–24.

If the handle end of a golf club shaft is clamped in a clamp that holdsthe shaft horizontally, then looking toward the tip of the distal end ofthe shaft, the shaft stiffness can be modeled, as shown in FIG. 1. Asseen in FIG. 1, shaft 10 can be considered as a mass m having twosprings of different spring constants k₁ and k₂ connecting it in twoorthogonal directions to two different surfaces 11, 12. If shaft 10 weresymmetrically stiff, then k₁ and k₂ would be equal. Normally, however,k₁ and k₂ are different. In fact, if one were to clamp the shaft inseveral different orientations, and each time measure the horizontal andvertical restoring forces, one might get different sets of values for k₁and k₂. The force F, as shown, is the force imposed to displace the tipof clamped shaft 10, e.g., to cause the tip to oscillate.

FIG. 2 shows the normalized horizontal and vertical displacement of thevibrating tip of shaft 10 as a function of time over two oscillationcycles, with horizontal displacement (x) represented by the solid line20 and vertical displacement (y) represented by the broken line 21,assuming the initial displacing force is imposed at an angle θ=40° tothe horizontal. FIG. 3 shows the same displacement of the tip of shaft10 as a phase plot 30, over two cycles, in x and y—i.e., FIG. 3 showstwo cycles of the path the tip follows as it would be seen by anobserver viewing the tip along the longitudinal axis of shaft 10,looking toward the handle end. FIG. 4 shows the phase plot 40 afterfourteen cycles. Analysis of these observed motions yields the locationof the planar oscillation plane—i.e., the angular orientation of shaft10 in which, if the initial displacing force F were applied along thatorientation, shaft 10 would oscillate substantially only along thatorientation, with the tip tracing back and forth substantially along aline.

As seen in FIG. 4, the phase plot 40 of the tip motion after asufficient number of cycles is substantially a rectangle. Theorientation of the planar oscillation plane is that of one of the twoorthogonal axes of that rectangle, where each axis of a rectangle isdefined as a line midway between, and parallel to, a respective pair ofsides of the rectangle. In the case of a true rectangle, it would besufficient to determine the orientations of the sides, as theorientations of the sides and the axes, according to the definition justset forth, are identical. However, the phase plot 40 of the tip motionof a golf shaft may not be a true rectangle, unless one observes aninfinite number of cycles, which is impractical because, first, it wouldnot be commercially acceptable and, second, the oscillations of the golfclub shaft ordinarily damp out before a true rectangle could beobserved. Therefore, the orientation of each of the two axes may becalculated by assuming that lines drawn through the four vertices of thequasi-rectangular shape of the phase plot are the diagonals of therectangle.

Having found the two axes of the rectangle, it is desirable to determinewhich one is the major axis, which may correspond to the principalplanar oscillation plane, and which is a minor axis—i.e., one of one ormore less stable planar oscillation planes. This can be determinedrigorously by measuring the oscillation frequencies along those twoaxes, as described below. The major axis would be expected to correspondto the principal planar oscillation plane if the shaft was caused tovibrate along a direction determined by measuring the load on thedeflected shaft as function of angle, and choosing the angle of maximumload as the direction in which to vibrate the shaft. It should be notedthat this “load test” could be carried out by clamping either the tip ordistal end, or the handle or proximal end, of the shaft, and measuringthe load as a function of angle with the unclamped end deflected.However, the subsequent steps of locating the planar oscillation planepreferably are carried out with the handle or proximal end clamped, andtherefore the load test preferably is carried out that way as well. Itshould also be noted that if the load test is not carried out, one mayfind a planar oscillation plane, but that planar oscillation planelikely will not be the principal planar oscillation plane.

FIG. 5 shows a plot 50 of tip oscillation as a function of time, with aseparate trace 51 for oscillations measured along the horizontal (x)axis and a separate trace 52 for oscillations measured along thevertical (y) axis. From these traces, frequency can be determined—e.g.,graphically by counting the positive-going zero crossings. However,these horizontal and vertical axes x and y are offset from the planaroscillation plane by an angle determined as described above. If thatangle is denoted θ, then the frequencies along these axes x and y asdetermined from the plot in FIG. 5 can be transformed into thecoordinate system of the golf club shaft, having axes x′ and y′ thatcorrespond to a stable planar oscillation plane and one of one or moreunstable planar oscillation planes, as follows, where f₁ is thefrequency at an angle θ from the x-axis—i.e., along the x′-axis, and f₂is the frequency at an angle θ from the y-axis (θ+90° from thex-axis)—i.e., along the y′-axis:$f_{1} = {\frac{f_{x}{f_{y}\left( {{{- f_{y}^{2}}\cos^{2}\theta} + {2f_{y}^{2}\cos^{4}\theta} - {3f_{x}^{2}\cos^{2}\theta} + {2f_{x}^{2}\cos^{4}\theta}} \right)}^{0.5}}{{f_{y}^{2}\cos^{2}\theta} + {f_{x}^{2}\cos^{2}\theta} - f_{x}^{2}}}$$f_{2} = {\frac{f_{x}{f_{y}\begin{pmatrix}{f_{y}^{2} - {3f_{y}^{2}\cos^{2}\theta} + {2f_{y}^{2}\cos^{4}\theta} -} \\{{f_{x}^{2}\cos^{2}\theta} + {2f_{x}^{2}\cos^{4}\theta}}\end{pmatrix}}^{0.5}}{{f_{y}^{2}\cos^{2}\theta} + {f_{x}^{2}\cos^{2}\theta} - f_{x}^{2}}}$If f₁ is greater than f₂, then one of the stable planar oscillationplanes of the golf club shaft is at an angle θ with respect to thex-axis. If f₁ is less than f₂, then one of the stable planar oscillationplanes of the golf club shaft is at an angle θ with respect to they-axis—i.e., θ+90° with respect to the x-axis. If the load test has beenperformed and used to determine the initial angle of vibration, then thestable planar oscillation plane so located can be expected to be theprincipal planar oscillation plane.

Although this mathematical technique, for determining which of theplanar oscillation planes already identified is the principal planaroscillation plane, is rigorous and precise, it does not include all ofthe parameters that may affect shaft oscillation. Therefore, in anotherpreferred embodiment of the invention, as described above and in moredetail below, the location of the principal planar oscillation plane islocated to a first-order approximation—i.e., at least to within thecorrect quadrant—by determining the orientation of the direction ofgreatest resistance to bending of the golf club shaft. This has thefurther benefit of quickly identifying the “hard” side of the principalplanar oscillation plane, as described above.

A first preferred embodiment of apparatus 60 for implementing thepresent invention is shown in FIGS. 6–13. Although apparatus 60 could bemade to implement the rigorous mathematics set forth above, it has beendetermined in practice that a simpler iterative process as describedbelow achieves acceptable results at lower cost. Therefore, in aparticularly preferred embodiment, apparatus 60 uses that simplerprocess.

In this preferred embodiment, apparatus 60 includes shaft testingassembly 70 and processing unit 61. Processing unit 61 can be any systemcapable of processing input data from sensors 74 and 77 of shaft testingassembly 70 and performing either the rigorous mathematical calculationsdescribed above or the simpler iterative calculations described below.As shown in FIG. 6, processor 61 is preferably a general purposecomputer such as a personal computer, which may, e.g., be based on aPENTIUM® central processing unit (CPU) 62 available from IntelCorporation, of Santa Clara, Calif., running a version of the WINDOWS®operating system available from Microsoft Corporation, of Redmond,Wash., and programmed with software as described below. However,processor 61 could also be hard-wired circuitry or one or moreprogrammed programmable logic devices dedicated to the functionsnecessary to locate the planar oscillation plane or planes of a golfclub shaft. In any event, processor 61 preferably also includes memory63 and mass storage 64, as well as interfaces for the sensors describedbelow.

Shaft testing assembly 70 preferably includes an elongated base 71,which is at least as long a golf club shaft. At one end of base 71 is ameasurement assembly 72, including a deflector assembly 73 and adeflection load sensor 74. At the other end of base 71 is a shaftholding and rotating assembly 75, including a rotatable chuck 76 forholding a golf club shaft. Apparatus 60 also includes a tip mass andsensor assembly 77 which during testing of a golf club shaft is mountedon the distal end of the golf club shaft and cooperates with deflectorassembly 73.

As seen in FIG. 8, shaft holding and rotating assembly 75 preferablyincludes rotatable chuck 76 which preferably may be conventional,preferably holding a golf club shaft by exerting radially inward forcesubstantially evenly around the shaft circumference. Chuck 76 preferablyis mounted at the end of axle 80, which preferably is journalled inbearings 81. Bearings 81 preferably are mounted on supports 82 so thatthe axis of rotation of axle 80, and by extension that of chuck 76 andthe golf club shaft being tested, is at a predetermined height abovebase 71. The end of axle 80 remote from chuck 76 preferably is connectedvia universal joint 83 to a potentiometer 84 that is used as an angularposition sensor as described below. Universal joint 83 prevents anyslight misalignment between the axis of axle 80 and the shaft ofpotentiometer 84 from damaging potentiometer 84. Similarly, a travelingnut 85 preferably is provided on axle 80 to act as a rotational stop tolimit rotation of axle 80 and thereby prevent damage that might resultfrom overrotation of potentiometer 84. An optional motor 86 may beprovided to rotate chuck 76, although manual rotation can also be used.In addition it is preferable to provide a clamp 87 to minimizevibrations of chuck 76 as it rotates. Clamp 87 preferably provides afriction fit to chuck 76 that is just light enough to allow rotation ofchuck 76. Screws 88 may be provided to adjust the jaw of clamp 87.

As seen in FIG. 9, measurement assembly 72 includes a base plate 90 thatis mounted to base 71. A load cell 91, such as a Model LCAE-2KG,available from Omega Engineering, Inc., of Stamford, Conn., is mountedto base plate 90, and a shaft tip restraining arm 92 is mounted to loadcell 91 on the side of load cell 91 opposite base plate 90, for apurpose to be described below. Measurement assembly 72 also preferablyincludes a deflector arm 93 pivotably mounted to base plate 90.Preferably, deflector arm 93 is mounted so that at least one side 930thereof is substantially perpendicular to base plate 90, and so that itpivots about an axis 94 that is substantially parallel to base plate 90.

Deflector arm 93 preferably has a projection 931, preferably extendingfrom side 930 thereof. Projection 931 preferably has a surface 932facing away from axis 94 that bears substantially the same angularrelationship to side 930 as does side 100 of tip mass and sensorassembly 77 to side 101 of tip mass and sensor assembly 77, for reasonsdescribed below.

As shown in FIG. 10, tip mass and sensor assembly 77 preferably has abody 102 with a mass of between about 190 grams and about 220 grams, andpreferably about 200 grams, to simulate the mass of a golf club head atthe distal end of a golf club shaft. In another embodiment, differenttip masses could be provided to more closely simulate different types ofclub heads, which have different masses. However, this latter embodimentwould be more costly, insofar as each different mass would need its ownset of transducers to collect displacement data, as well as differentcomputations based on those data.

The presence of body 102 on the end of a golf club shaft when the shaftis deflected and allowed to oscillate during testing in accordance withthe present invention, as described below, not only mimics the effect ofa club head during a swing, but also provides “reaction mass” thatprevents the shaft oscillations from damping out before sufficient datacan be collected. The transducers that collect the displacement datapreferably are two accelerometers 103, 104—such as Model 8303A availablefrom Kistler Instrument Corp. of Amherst, N.Y.—aligned along twodifferent axes. Preferably, the two axes are orthogonal to one another,but that is not necessary; as long as the angular relationship betweenthe axes is known, the motion recorded by accelerometers 103, 104 can beresolved computationally into two orthogonal components. Alsopreferably, the two axes are parallel and perpendicular, respectively,to base 71. Again, however, that is not necessary.

Tip mass and sensor assembly 77 preferably has an attachment structurefor attaching to the tip of a golf club shaft. Preferably, theattachment structure includes a bore 105, slightly larger in diameterthan an average golf club shaft, in body 102, into which the shaft maybe introduced, and a set screw 106 for tightening body 102 onto theshaft. Alternatively, some sort of quick-release clamp can be provided,particularly for use in an automated system as described below.

In addition, body 102 could be divided by a plane or other surfacepassing through bore 105, so that it can be assembled around a shaftinstead of being slipped over the shaft tip. This is particularly usefulwhen analyzing the shaft of a pre-existing golf club and it is desirednot to remove the club head from the shaft. The two portions (not shown)of body 102 could be fastened together, after being assembled around theshaft, by any suitable clamps or other fasteners. For example, the twoportions could be hinged at one edge of the dividing surface, with oneor more fasteners being provided on the opposite edge.

As discussed above, there preferably is the same relationship betweenthe orientations of sides 100, 101 of tip mass and sensor assembly 77 asthere is between surfaces 930, 932 of deflector arm 93. This is so thattip mass and sensor assembly 77 can be repeatedly lined up the same wayfor every test, by resting sides 100, 101 against surfaces 930, 932.

In order to test a golf club shaft, the shaft 110 is mounted in chuck 76as shown in FIG. 11. The tip, or distal end, of shaft 110 is thendeflected and restrained under the lip 120 of shaft tip restraining arm92, as shown in phantom in FIG. 11, so that the restoring force tendingto straighten shaft 110 can be measured by load cell 91. Chuck 76 isthen rotated—manually, or by motor 86 preferably under control ofprocessor 61—while the restoring force is recorded by computer 61 as afunction of angle, which is determined by potentiometer 84, to which aknown voltage is applied. By well-known voltage divider techniques, thechanging resistance is translated to a changing voltage, which can beconverted to an angle.

It might be expected that when the upward restoring force is a maximum,then the point of maximum asymmetry of the shaft, representing the hardside of the principal planar oscillation plane, is facing upward. It hasbeen found empirically, however, that that is not so, but that the hardside is within the quadrant that is facing upward when the maximum forceis measured. The angle of the maximum force is therefore recorded inthis static portion of the test, and the remainder of the test, which isdynamic, is conducted.

In the dynamic portion of the test, the tip or distal end of golf shaft110 is oscillated with tip mass and sensor assembly 77 in place. Whilein the static portion of the test the tip preferably is deflectedvertically, in the dynamic portion of the test the deflection ispreferably horizontal, although any direction can be used in eitherportion of the test. The reason for preferring horizontal deflection inthe dynamic portion of the test is that, first, the effect, on theresults, of gravity acting on the tip mass is minimized, and, second, itis easier to oscillate the shaft without it hitting base 71. Therefore,before the dynamic portion of the test is initiated, chuck 76 preferablyis rotated about 90°, so that the estimated orientation of the principalplanar oscillation plane, which had been vertical, is now horizontal.

In the apparatus so far described, tip mass and sensor assembly 77 isapplied, and a horizontal impulse is imparted, to golf club shaft 110,as follows. With the proximal or handle end 111 of golf club shaft 110held in chuck 76, and deflector arm 93 standing erect, bore 105 in body102 of tip mass and sensor assembly 77 is placed over distal or tip end112 of golf club shaft 110. Tip mass and sensor assembly 77 is thenmanipulated until surfaces 100, 101 of body 102 are firmly seatedagainst surfaces 930, 932 of deflector arm 93, placing accelerometers103, 104 in their predetermined desired orientations. A portion ofsurface 100 not occupied by accelerometer 103 is used for this purpose,so that accelerometer 103 does not interfere with the seating of body102. Although accelerometers 103, 104 are shown connected to processor61 by wires 62, a wireless connection (not shown) could be provided.

A preferably substantially horizontal impulse is provided to tip massand sensor assembly 77 by deflecting tip 112 of golf club shaft 110 toside 120 of deflector arm 93 opposite side 930, as seen in FIG. 12, andthen, preferably in a sudden motion, pivoting deflector arm 93 out ofits erect position, allowing the restoring force in deflected golf clubshaft 110 to provide a horizontal impulse to start tip 112 of golf clubshaft 110 to begin vibrating, along with tip mass and sensor assembly77, in the manner described above in connection with FIGS. 2–5.

Although the initial deflection of golf club shaft 110 behind deflectorarm 93, as well as the pivoting of deflector arm 93 to allow tip 112 tooscillate, can be accomplished manually, they can also be accomplishedautomatically. Thus, an arm 121 bearing a finger 122, driven by a motor123 through suitable gearing or linkage 124 that provides the necessaryhorizontal and vertical components of motion, can be used to move tip112 of golf club shaft 110 from its neutral position 1200 to theposition behind deflector arm 93. This could involve both vertical andhorizontal movement of tip 110 by finger 122, or finger 122 could movesolely horizontally while motor 125 pivots deflector arm 93 out of theway temporarily and then restores deflector arm 93 to the erectposition. Similarly, the pivoting of deflector arm 93 to allowoscillation to begin can be performed by motor 125 instead of manually.

As a further alternative, instead of applying an impulse by deflectingshaft 110 behind deflector arm 93 and then releasing arm 93, ahorizontal plunger or ram (not shown) could be used to strike tip massand sensor assembly 77 rapidly and for a short time.

Each of accelerometers 103, 104 records acceleration in one of tworespective directions, which preferably are orthogonal to one another,and preferably are horizontal and vertical, respectively. However, anytwo directions may be used, as long as they are known, and thehorizontal and vertical components can be calculated. The accelerationsmay be integrated twice over time to determine horizontal and verticaldisplacements, but the acceleration is generally indicative of thedisplacement and may be used directly, saving computational resourcesand time that would be needed to perform the integrations.Alternatively, displacement can be measured directly, for example, byproviding, instead of accelerometers 103, 104, a light source, such as alaser or light-emitting diode (not shown), on the end of tip mass andsensor assembly 77 emitting light along the direction of thelongitudinal axis of golf club shaft 110. A light sensitive detectorarray (also not shown) could be placed substantially perpendicular tothe emitted light beam, which would trace the displacement of tip 112 onthe detector array, recording the displacement directly. Regardless ofhow the data are collected, they can be plotted as a function of timeand used to derive displacement and frequency data that are then used,as described above, to mathematically determine the preferred angularorientation in which lies the principal planar oscillation plane. Thedirection of the principal planar oscillation plane closer to theestimated orientation determined by load cell 91 would be considered the“hard” side of the principal planar oscillation plane of golf club shaft110, which preferably should be aligned perpendicular to, and facing, orin any other predetermined orientation with respect to, the club headface. However, the load cell test could be eliminated, insofar asaligning golf club shaft 110 with a planar oscillation plane in adesired orientation with respect to the club head face, whether the hardside of that planar oscillation plane faces toward or away from theface, may be better than having that planar oscillation plane at arandom orientation relative to the club head face, and also insofar asaligning any planar oscillation plane with respect to the club headface, even if it is not the principal planar oscillation plane, may bebetter than a random orientation. It should be remembered, however, thatif a random planar oscillation plane, rather than the principal planaroscillation plane, is found for each golf club shaft in a set, then evenif the planar oscillation plane so found for each shaft is orientedsimilarly relative to its respective club head, the set cannot beassumed to be uniformly oriented.

Once the location of the desired planar oscillation plane (preferablythe principal planar oscillation plane) has been determined, shaft 110preferably is marked to indicate the orientation of that planaroscillation plane. Marking may be accomplished by applying a pigment(e.g., paint or ink) to the surface of shaft 110. For example, an inkmarker 130 having a marking tip 131 could be mounted on a frame 132 asshown in FIG. 13. After the preferred orientation has been determined,shaft 110 can be rotated so that the preferred orientation is alignedwith marking tip 131, which then applies a mark to shaft 110.Alternatively, 130 could represent a paint reservoir, while 131 wouldrepresent a paintbrush or a jet of sprayed paint. As a furtheralternative, marking of shaft 110 could be accomplished using a directedenergy beam or a particle beam to etch a marking into the surface ofshaft 110. In such an alternative, 130 could represent a high-energylaser, while 131 would represent the laser beam, or 130 could representan electron gun while 131 would represent the electron beam. Optionally,either shaft 110 or the marking assembly could be moved parallel to theshaft longitudinal axis during marking so that the marking on the shaftis a line instead of a dot, to increase its visibility. Alternatively,as discussed above, a marked label, such as a sticker or decal, bearingalignment markings, can be applied to shaft 110.

The preferred method 140 according to the invention for locating thepreferred orientation (i.e., either any planar oscillation plane or theprincipal planar oscillation plane), using apparatus 60, is diagramed inFIGS. 14–17. Method 140 preferably starts with load test 141, describedabove, which uses load cell 91 to estimate the orientation of theprincipal planar oscillation plane and which at least identifies whichof the two sides of the principal planar oscillation plane is the “hard”side of the planar oscillation plane, by measuring the restoring forceas a function of angle of a deflected shaft that is rotated through atleast 360°. Load test 141 could be omitted, but only if one is preparedto find any planar oscillation plane, rather than the principal planaroscillation plane in particular (unless another technique is used toidentify the principal planar oscillation plane). Where load test 141 isperformed, the result is used as a starting point for planar oscillationplane location step 143, below. Alternatively, load test 141 could beperformed on a stand-alone basis to measure the symmetry of a shaft.

After load test 141 is performed, optional “logo up” test 142 isperformed, to gather data, for comparison purposes, regarding theoscillation of a golf club shaft in its factory installed orientation.Conventional golf clubs are typically assembled with the manufacturer'slogo, which is printed on the shaft, facing toward the club head face,in what is referred to as a “logo up” configuration. Some manufacturersalign the logo 180° away from the club head face in a “logo down”configuration, or in other configurations. During “logo up” test 142,the shaft is positioned in its original factory installed position, buttest 142 is referred to as the “logo up” test because most frequentlythe factory position has the logo facing upward. In any case, becausethe logo is printed at a random location on the shaftcircumference—i.e., without the benefit of knowing the location of anyplanar oscillation plane—the factory alignment is purely randomregardless of the actual logo position.

As described above, planar oscillation plane location procedure 143 isperformed next. After procedure 143 has been performed, an optionalreport printing step 144, in which some or all of various parametersregarding the golf shaft whose preferred orientation has been found areprinted. Finally, in an optional save step 145, various of the dataacquired during steps 141–144 are saved (e.g., in mass storage 64).

Load test 141 is shown in more detail in FIG. 15. At step 150, a golfclub shaft 110, which may have been removed from a golf club, is placedin chuck 76 at an arbitrary starting angle. Tip 112 of golf club shaft110 is deflected and restrained under shaft tip restraining arm 92 sothat the restoring force in the deflected shaft 110 is measured by loadcell 91. The shaft can be deflected and secured manually, or thedeflection and securing can be accomplished automatically. Thus, an arm126 bearing a finger 127, driven by a motor 128, through suitablegearing or linkage 129 that provides the necessary horizontal andvertical components of motion, can be used to move tip 112 of golf clubshaft 110 from its neutral position 1200 to position 1201 under shafttip retention arm 92.

Once tip 112 is under shaft tip retention arm 92, then in step 151 chuck76 preferably is rotated about 200° in one direction (which may bedesignated the negative rotation direction). Next, at step 152, chuck 76is rotated at least 360° in the opposite direction (which may bedesignated the positive rotation direction) while data is acquired fromload cell 91 and recorded as a function of angle. Preferably, in step152, chuck 76 is rotated about 400° and 40° (preferably the first andlast 20°) is discarded. Alternatively, however, the reverse rotation ofstep 151 may be omitted, as long as data are recorded through at least360°, and if data are recorded through more than 360°, then any amountof rotation greater than 360° may be used and any portion—all at thebeginning, all at the end, or any combination of beginning and end—maybe discarded to provide 360° worth of data.

At step 153, the data gathered in step 152 are examined, and the angle Acorresponding to the maximum load measured by load cell 91 isdetermined. If desired, the load as a function of angle may be graphedfor display. Next, at step 154, the start angle S, for use in planaroscillation plane location test 143, is set to A-90°. This takes intoaccount the change of orientation from vertical to horizontal as betweenthe load test 141 and the planar oscillation plane location test 143, asdescribed above.

After the conclusion of load test 141, “logo up” test 142, shown indetail in FIG. 16, may be conducted. The purpose of “logo up” test isprimarily to provide a “before” comparison to the “after” result to beobtained after performing planar oscillation plane location test 143.Therefore, as stated above, “logo up” test 142 is optional. Inparticular, while “logo up” test 142 may be used primarily as apromotional tool in an aftermarket situation—i.e., by a golf clubretrofitter—to show the improvement obtained by realigning the shaft ofa golf club in accordance with the invention, it probably would not beused by a golf club manufacturer who produces “spine-aligned” golfclubs, because there is no need to show comparative data.

“Logo up” test 142 begins at step 160 where golf club shaft 110, which,again, may have been removed from a golf club, is placed in chuck 76. Ifit had previously been part of a completed golf club, shaft 110 isplaced in chuck 76 in the same orientation in which it was oriented inthe golf club, as the club would have been positioned by a golferadjacent a ball before the start of the golfer's swing. In most cases,this would be with the manufacturer's logo facing up, but sometimes thelogo faces down or in a random direction. If test 142 is being performedon a golf club shaft that has never been part of a golf club, thenpreferably it is tested with its logo up, or with the logo in whateverposition, whether or not it is up, is recommended by the shaftmanufacturer for alignment of the logo when assembling a golf club. Tipmass and sensor assembly 77 is then mounted on tip 112 of shaft 110.

Next, at step 161, an impulse is applied to tip mass and sensor assembly77 in one of the ways described above and orthogonal—preferably,horizontal and vertical—acceleration data are gathered, preferably forabout 4 seconds. These data preferably are integrated at step 162 toyield orthogonal—preferably horizontal and vertical—displacement data asfunctions of time, which preferably are saved at step 163 for latercomparison with the results after alignment of shaft 110, and the datapreferably also are graphed at step 163 for display to the owner of thegolf club of which shaft 110 is a part. The maximum out-of-planedisplacement—i.e., preferably the maximum verticaldisplacement—preferably is also saved at step 163 for display to theowner. Test 142 is now complete.

The system next proceeds to planar oscillation plane location test 143.As shown in FIG. 17, test 143 starts at step 170 where a counter J isinitialized to zero. Next, at step 171, chuck 76, still holding shaft110, is rotated to the start angle S previously computed. If no startangle S has been computed, test 143 starts at an arbitrary angle.

At step 172, if tip mass and sensor assembly 77 has not previously beenattached to tip 112 it is attached, and in any case an impulse isapplied to tip mass and sensor assembly 77 in one of the ways describedabove and orthogonal—preferably, horizontal and vertical—accelerationdata are gathered, preferably for about 4 seconds. These data preferablyare integrated at step 173 to yield orthogonal—preferably horizontal andvertical—displacement data as functions of time. At step 174, thecounter J is incremented by one. At test 175, the system tests to see ifJ=1. If, as on this first pass, J=1, then the system skips directly tostep 177.

At step 177, the system sets a variable YMAX(J) equal to the maximum outof plane deviation value from step 173. The system then proceeds to test178 where it determines if J=1, meaning it is the first pass through theloop. There preferably are always at least three passes through theloop. If at test 178 J=1, then at step 179 the angle S is incremented by10°. At step 1700, in order to keep S between +180° and −180°, ifS>180°, then S is set to S-360°. Next, at step 1701, the frequencies ofthe horizontal and vertical oscillations are determined; this may bedone from the displacement-vs.-time data from step 173. Frequency dataare commonly used to measure the stiffness of golf club shafts, andthese data are useful for comparison. Note, however, that the frequencyof shaft oscillation is dependent on the length of shaft protruding fromwhatever holding device is used and on the characteristics of theholding device (e.g., the length and tightness). Therefore, if anycomparison is to be made, care should be taken to use the same holdingdevice and to assure that the same length of shaft is free to vibrate.

After step 1701, the system loops back to step 172, and steps 172–174are carried out again. This time, at test 175 J≠1, and at step 176 thedata from step 173 are saved along with angle S, and the system proceedsto step 177. Again at step 177 variable YMAX(J) is set equal to themaximum out of plane deviation value from step 173. This time at test178 J≠1, and the system proceed to test 1702 to determine if J=2. Onthis second pass, J=2 and the system proceeds to test 1703 to determineif YMAX(J)>YMAX(J−1). If not, that means in this iteration theout-of-plane excursions are smaller, meaning the angle S is closer tothe preferred orientation—i.e., to the planar oscillation plane—and atstep 1704 the variable SIGN is set to +1, the variable Y is set to thevalue of YMAX(J), and the variable AMP is set to 1.0, and the systemproceeds to step 1706. If at test 1703 YMAX(J)>YMAX(J−1), that means inthis iteration the out-of-plane excursions are larger, meaning the angleS is further from the planar oscillation plane, and at step 1705 thevariable SIGN is set to −1, the variable S(J) is set to the value ofS(J−1), the variable YMAX(J) is set to the value of the variableYMAX(J−1) and the variable Y is then set to the value of YMAX(J), andthe variable AMP is again set to 1.0, and the system proceeds to step1706. Note that in either step 1704 or step 1705, AMP can be set to alower value to cause the result to converge sooner, but with loweraccuracy, while setting AMP higher increases accuracy but increases thenumber of iterations before convergence. This is a trade-off betweenspeed and accuracy.

At step 1706 the system calculates the variable POP=SIGN(45−(90/π)cos⁻¹(Y/AMP)), and at step 1707 the value of S is set to S+POP. At step 1708,in order to keep S between +180° and −180°, if S>180°, then S is set toS−360°. Similarly, at step 1709, in order to keep S between +180° and−180°, if S<−180°, then S is set to S+360°. The system then returns tostep 1701 to calculate the frequencies, and once again loops back tostep 172. This time, on the third pass, at test 178 J≠1, and at test1702 J≠2, and the system advances to test 1710 to determine ifYMAX(J)>YMAX(J−1). If it is, then the values are converging, and thesystem proceeds to test 1711 to determine if the out-of-plane excursionon the last iteration (YMAX(J−1)) is less than the maximum out-of-planeexcursion during the “logo up” test 142. If it is, then the currentorientation is the preferred orientation, and at step 1712 the variablePOP, representing the preferred orientation, is set to the value of thevariable S, representing the current orientation. At step 1713, theshaft frequencies are again calculated as in step 1701, and test 143ends at 1714.

If at test 1711, the out-of-plane excursion on the last iteration(YMAX(J−1)) is not less than the maximum out-of-plane excursion duringthe “logo up” test 142, then at step 1715, the variable POP,representing the preferred orientation, is set to the “logo up” angle.At step 1713, the shaft frequencies are again calculated as in step1701, and test 143 ends at 1714.

If at test 1710, YMAX(J)≯YMAX(J−1), then the values have not converged,then at step 1716, Y is set to the value of YMAX(J). The system thenrecalculates POP at step 1706 and from there goes through the loop atleast one more time.

If optional “logo up” test 142 is not performed, then if test 1710indicates convergence, test 1711 is not performed and the systemproceeds directly from test 1710 to step 1712.

After completing planar oscillation plane location test 143, the systemproceeds to report printing step 144 where the values of the followingdata preferably are printed (and determined if necessary): load as afunction of angle (as determined in load test 141); load symmetry index(LSI), which is a measure of the variability in stiffness of the shaft(LSI=100(1−((P_(max)−P_(min))/P_(max))), where P_(max) and P_(min) arethe maximum and minimum loads, respectively, measured in step 152);displacement plot at the “logo up” angle; displacement plot at the POPangle; displacement as a function of time at the “logo up” angle and the“hard” and “soft” POP angles (the latter two should be exactly180°apart); the horizontal and vertical frequencies and the maximumout-of-plane excursions at the “logo up” and POP angles; and a frequencyindex equal to the ratio of the horizontal frequency at the POP angle tothe horizontal frequency at the “logo up” angle, which is a comparativemeasure, in the form of a percentage improvement, of stiffness in thehit direction as between the original “logo up” configuration of thegolf club and the aligned configuration.

Next at step 145 the data are saved. In a full save, all data are saved.There preferably is also a “quick save” in which all the data printed instep 144 are saved except for the complete load-vs.-angle data and thecomplete displacement data at the “logo up” and POP angles. Followingsaving step 145, process 140 ends at 146.

An alternative embodiment of apparatus 1870 for determining theprincipal planar oscillation plane of a golf club shaft, and anassociated method, are now described in connection with FIGS. 18–22.

Apparatus 1870 is substantially fully automated. Substantially the onlysteps performed manually in using apparatus 1870 are the adjustment ofthe location of instrumentation table 1872 to conform to the length ofthe golf club shaft 110 being measured, the mounting of shaft 110 inchuck 1876, and the mounting of tip mass and sensor assembly 1877 onshaft 110.

Shaft testing assembly 1870 preferably includes an elongated base 1871,which is at least as long as the longest golf club shaft expected to betested. At one end of base 1871 is a measurement instrumentation table1872, which can be translated along base 1871 to accommodate golf clubshafts of different lengths. Preferably, instrumentation table 1872 hasa base 1890 with downward projections (not shown) that ride in slot 1891in base 1871, as well as a roller 1892 that rides on supporting surface1800 of enclosure 1801. Screw 1874 preferably is provided to lockinstrumentation table 1872 in a selected position.

Instrumentation table 1872 includes deflector/deflection load sensorassembly 1878, used to determine straightness. Instrumentation table1872 also includes vibration initiator assembly 1873, used to initiatevibration of shaft 110 for determination of stiffness and to locate theplanar oscillation plane of shaft 110, and vibration dampener assembly1897 whose function is explained below.

At the other end of base 1871 is a shaft holding and rotating assembly1875, including a rotatable chuck 1876 for holding a golf club shaft110.

As seen in FIG. 18, shaft holding and rotating assembly 1875 preferablyincludes rotatable chuck 1876 which preferably may be conventional,preferably holding a golf club shaft by exerting radially inward forcesubstantially evenly around the shaft circumference. Chuck 1876preferably is mounted at the end of axle 1880, which preferably isjournalled in bearings 1881. Bearings 1881 preferably are mounted onsupports 1882 so that the axis of rotation of axle 1880, and byextension that of chuck 1876 and the golf club shaft being tested, is ata predetermined height above base 1871. Mounted at the end of axle 1880remote from chuck 1876 preferably is a toothed pulley 1883, connected bya toothed belt 180 to a similar toothed pulley 1884 of servo motor 1885whose angular position can be controlled with precision by processor 61,such as a Model SM2315 smart motor available from Animatics Corporation,of Santa Clara, Calif. Motor 1885 is preferably mounted under supports1882. Preferably, the space under supports 1882 is also used (not shown)as a junction box for the various sensors and other electrical andelectronic components described below. Also preferably, the space undersupports 1882 is enclosed, as by acrylic panels (not shown), to keep outdust and dirt and to prevent users from coming into contact with anyexposed electrical connections.

Instrumentation table 1872 is shown in more detail in FIGS. 19 and 20.Deflector/deflection load sensor assembly 1878 includes a verticallyextending bar 190 mounted so that golf club shaft 110 passes above, butspaced from, its upper end. Bar 190 can be moved vertically, andpreferably an actuator such as pneumatic cylinder 191 is provided tomove bar 190 upwards to that its upper end engages shaft 110 anddeflects shaft 110 upward by a predetermined amount. Any other suitableactuator can be used, including a linear actuator such as a solenoid orhydraulic cylinder, or rotary actuator such as a motor. Preferably, theupper end of bar 190 is provided with a seat, such as a V-shaped seat193, to engage shaft 110. A compressive load sensor 192 such as Model9222 from Kistler Instrument Corp., of Amherst, N.Y., is providedbeneath cylinder 191 to measure the restoring force exerted by shaft 110when it is deflected upward by bar 190. As discussed above in connectionwith the previous embodiment, because the load test measurements aretaken vertically while the planar oscillation plane locationmeasurements are taken horizontally, load test data are recorded atangles offset 90° from the angles at which they are measured.

Vibration initiator assembly 1873 of instrumentation table 1872preferably includes electromagnet 1894 preferably mounted for horizontalmovement transverse to the longitudinal axis of shaft 110, preferablyunder the influence of pneumatic cylinder 1895 mounted on support 1896.Any other suitable actuator can be used, including a linear actuatorsuch as a solenoid or hydraulic cylinder, or a rotary actuator such as amotor.

Apparatus 1870 preferably also includes proximal end shaft markingmechanism 1887 and distal end shaft marking mechanism 1886 for markingon golf club shaft 110 the location of the principal planar oscillationplane once determined. Distal end shaft marking mechanism 1886preferably includes one or more (e.g., two, as shown) pens 1888 to makeone or more (e.g., two) marks at the distal end of shaft 110 which canbe used to align shaft 110 with a golf club head in the desiredorientation, while proximal end shaft marking mechanism 1886 preferablyincludes one or more pens 1888 to make one or more marks at the proximalend of shaft 110. In both shaft marking mechanisms, a respectivepneumatic cylinder preferably is used to raise pens 1888 into contactwith shaft 110. Preferably, to steady shaft 110 for the marking process,shaft deflector bar 190 is raised by cylinder or other actuator 191 todeflect shaft 110 vertically and, more important, to hold it from movinghorizontally during the marking process. Another cylinder or otheractuator 194 then actuates pens 1888 of distal end shaft markingmechanism 1886. Cylinder or other actuator 1889 of proximal end shaftmarking mechanism 1887 is also actuated so that pen 1888 at the proximalend mark shaft 110, preferably while bar 190 continues to deflect andhold shaft 110. Alternatively, as discussed above, one or more stickersor decals, bearing alignment markings, may be applied to shaft 110. Themarks made on shaft 110 are made primarily to align shaft 110 with thegolf club head, and therefore the number and location of the marks is amatter of choice, based on the needs of the equipment used to attach thehead to shaft 110, or the needs of an individual attaching the head toshaft 110 manually.

Tip mass and sensor assembly 1877 is similar to assembly 77 above. Tostart the process, the user mounts tip mass and sensor assembly 1877,using the face of electromagnet 1894 as an alignment surface after golfclub shaft 110 is inserted into chuck 1876. Cylinder or other actuator1895 is set so that in its rest position, it presents electromagnet 1894in the correct position to be used as an alignment surface for themounting of tip mass and sensor assembly on shaft 110 with shaft 110 inits neutral position. After tip mass and sensor assembly 1877 has beenmounted and aligned, electromagnet 1894 is energized. Cylinder or otheractuator 1895 is then actuated to withdraw electromagnet 1894 in adirection away from the longitudinal axis of shaft 110 to deflect shaft110 horizontally. At or before the end of the travel of electromagnet1894 in the withdrawal direction, electromagnet 1894 is de-energized,releasing its hold on assembly 1877, causing shaft 110 to oscillatesubstantially horizontally.

As shaft 110 oscillates, the motion of the shaft tip as sensed by tipmass and sensor assembly 1877 is recorded by processor 61, and inparticular the maximum out-of-plane vertical acceleration ordisplacement and the vibration frequency are noted.

Preferably, after enough data have been gathered, a dampener 1898, suchas a foam pad, is moved into engagement with shaft 110, preferably bycylinder 1899 mounted on support 1900, to stop the shaft oscillation.

Whether or not dampener assembly 1897 is provided, electromagnet 1894 isnext re-engaged with assembly 1877. While servo motor 1885 rotates shaft110 to the next angular position, which preferably is 10° from thecurrent position, engagement with the face of electromagnet 1894 keepsassembly 1877 aligned during shaft rotation. Set screw 106 preferablyhas a nylon tip so that if assembly 1877 is being held from rotating byelectromagnet 1894, shaft 110 can nevertheless rotate relative toassembly 1877. Electromagnet 1894 is or remains energized and is againwithdrawn to deflect shaft 110 which again preferably is released tovibrate by de-energizing electromagnet 1894. At this new angularposition, the displacement data, including the maximum out-of-planeexcursion of the shaft tip, and the frequency data are again recorded.This is repeated at preferably uniform angular intervals, preferablyeach 10°, so that out-of-plane excursion data and frequency data areavailable for thirty-six angular positions. Although the angularintervals are preferably uniform, the time spent at each angularposition may not be equal. For example, in a preferred embodiment, moredata may be taken at the “logo up” position and at the principal planaroscillation plane position to provide more detail for graphical display(see below).

As an alternative to tip mass and sensor assembly 1877, a tip massassembly 261, shown in FIG. 26, may be used. Tip mass assembly 261 issimilar in size and mass to tip mass and sensor assembly 1877, exceptthat it does not include accelerometers or any other sensors,eliminating the need for a wired or wireless connection to processor 61.Tip mass assembly 261 includes a flat plate 262 for interaction withelectromagnet 1894. For reasons discussed below, plate 262 is preferablymounted at an angle of 45° relative to the faces of tip mass assembly261.

For use in conjunction with tip mass assembly 261, apparatus 1870preferably is equipped with a pair of laser distance sensors 263, 264,each of which may be a Type OADM laser distance sensor which isavailable from Baumer Electric AG, of Frauenfeld, Switzerland. As shownin FIGS. 27 and 28, sensors 263, 264 preferably are mounted so that whentip mass assembly 261 is mounted on a shaft 110, sensors 263, 264 are onthe opposite side of tip mass assembly 261 from electromagnet 1894. Morepreferably, upper sensor 263 is mounted at a height such that its beamimpacts substantially at the center of side 270, while lower sensor 264is mounted at a height such that its beam impacts substantially at thecenter of side 271, when tip mass assembly 261 is mounted with plate 262substantially vertical. It should be noted that set screw 106 is shownin phantom in FIGS. 27 and 28 because while it is located in side 270,it is further down side 270 along the longitudinal axis of shaft 110than is the point at which the beam 272 from sensor 263 intersects side270. Therefore, set screw 106 does not interfere with the operation ofsensor 263.

Each of sensors 263, 264 includes a laser source and a photodetector andoperates by measuring the time it takes for a laser pulse to reach asurface of tip mass assembly 261 and return to the photodetector. As isapparent from FIG. 27, if tip mass assembly vibrates purelyhorizontally, the distance d_(U) from upper sensor 263 to side 270 willalways be substantially equal to the distance d_(L) from lower sensor264 to side 271. However, as can be seen from FIG. 28, if there is anyvertical component in the vibration of tip mass assembly 261, then, evenif there is no horizontal component of vibration, the two distancesd_(U) and d_(L) will differ.

Assuming:

(1) the horizontal and vertical displacements of tip of shaft 110 aredenoted x and y, respectively;

(2) the difference between d_(U) as measured when tip mass assembly 261is at rest and d_(U) as measured during a particular measurement isdenoted x_(U); and

(3) the difference between d_(L) as measured when tip mass assembly 261is at rest and d_(L) as measured during a particular measurement isdenoted x_(L);

then x and y can be derived from x_(U) and x_(L) as follows:x=(x _(L) +x _(U))/2y=(x _(L) −x _(U))/2

It will be apparent from the geometry that tip mass assembly 261 couldbe mounted so that sides 270 and 271 are not at 45° angles relative tovertical (or relative to the horizontal vibration plane), but rather atsome other angles oblique to the vertical (or to that plane), but thenthe mathematics for deriving x and y from x_(L) and x_(U) would besignificantly more complicated. It will also be apparent that it is notnecessary for detectors 263, 264 to be mounted opposite the respectivemidpoints of sides 270, 271, as long as the beam of detector 263 willintersect side 270 and the beam of detector 264 will intersect side 271as tip mass assembly 261 vibrates. However, if other mounting positionsof sensors 263, 264 are used, care should be taken that the positionschosen are not ones such that the aforementioned condition could beviolated by a degree of vibration within the expected range of vibrationof tip mass assembly 261.

It will also be appreciated that sides 270, 271 of tip mass assembly 261should not be perfectly reflective. If sides 270, 271 were perfectlyreflective, all of the laser energy emitted by the laser sources insensors 263, 264 would be reflected away from the detectors in thosesensors. There must be sufficient specular reflection that some of thelaser energy returns to its source. Preferably, the surfaces of sides270, 271 are made as close to a “white paper surface”—i.e., a surfacethat, when excited by laser energy, re-emits omnidirectionally at thesame wavelength as the incident beam—as possible. The two sensors 263,264 should be far enough apart that reflected or re-emitted energy fromside 270 does not reach the detector of sensor 264, and that reflectedor re-emitted energy from side 271 does not reach the detector of sensor263. Alternatively, the two sensors could operate at differentwavelengths, so that the signal of one sensor could not be read by theother sensor.

The use of sensors 263, 264 instead of accelerometers 103, 104 providesdisplacement data directly, without the need for integration ofacceleration data. However, as stated above, for purposes of thisinvention, acceleration measurements and displacement measurements yieldthe same results.

After the completion of all measurements at all angular positions,assembly 1877 or 261 is then removed manually from shaft 110.

The actuation of servo motor 1885 and the various cylinders/actuators ispreferably automated under the control of processor 61, so that themultiple measurements can be taken quickly. If measurements are taken atevery 10°, the full series of measurements preferably is completedwithin about two minutes or less, and preferably within about 30seconds. If pneumatic cylinders are used as the actuators, the variouspneumatic cylinders preferably are powered by compressor 2000, whichpreferably is located within enclosure 1801 and is connected to thevarious cylinders by hoses 2001. Enclosure 1801 may be used to houseother components (not shown) of apparatus 1870.

The result of the measurements is a tabulation, for each angularposition, of tip position (particularly out-of-plane displacement) andvibration frequency. To locate the principal planar oscillation plane,the out-of-plane displacement may be plotted in polar coordinates as afunction of angle. An example of such a plot is shown in FIG. 21. Ateach angular position, the distance of the curve from the originrepresents the out-of-plane displacement at that angle. A typical golfclub shaft will have a multi-lobed plot such as that shown in FIG. 21,although the number of lobes may vary among different shafts. The cusps210 between the lobes, where the plotted curve approaches closer to theorigin, are the local minima of out-of-plane displacement. Except forvery anomalous shafts, the number of cusps 210 is expected to be even,and each cusp 210 at a particular angle should have a mate 180° awayfrom it. Each such pair (indicated by dashed lines 211) represents oneof the planar oscillation planes of the shaft, with the principal planaroscillation plane ordinarily being represented by that pair of cuspsthat are closest to the origin. It should be noted that by plotting theobserved data graphically, the principal planar oscillation plane can beprecisely located even if its position is not one of the angularpositions at which measurements actually were taken. Processor 61preferably is programmed with software to plot the data and select theprincipal planar oscillation plane automatically.

In a preferred embodiment, such software fits curve 212 to data points213 using a Fourier series approximation. If, for each point 213 takenat a particular angle θ, the distance from the origin is indicated asr(θ), then points 213 can be fit to the following series:r(θ)=A ₀ +A ₁ cos θ+B ₁ sin θ+A ₂ cos (2θ)+B ₂ sin (2θ)+ . . . +A _(m)cos(mθ)+B_(m) sin(mθ)where the closeness of the fit improves as the number of terms, m,increases. However, where N is the number of data points, the number ofterms is limited:m<(N−1)/2.Thus, in the preferred embodiment, where the number of points is 36, themaximum number of terms is 17.

In the series above, the coefficients are defined as follows:$A_{0} = {\left( {1/N} \right){\sum\limits_{\theta}r}}$(i.e., the average distance of points 221 from the origin);${A_{j} = {\left( {2/N} \right){\sum\limits_{\theta}{y\;{\cos({j\theta})}}}}};{and}$$B_{j} = {\left( {2/N} \right){\sum\limits_{\theta}{y\;{{\sin({j\theta})}.}}}}$

It has been observed empirically that a minimum of four terms (m=4) isneeded for an acceptable curve fit. The fit improves as m increases.

Once curve 212 is fit to points 213, the first derivative is taken andset equal to zero to find the maxima (tips of the lobes) and the minima(cusps 210). The second derivative is then taken at each extremum toidentify which is a minimum or cusp (positive second derivative) andwhich is a maximum (negative second derivative). It has been found thatfor curve fitting purposes, thirty-six data points (0° to 350°) shouldbe used if the end points (0° and 360°) are to have the same slope anddisplacement, but thirty-seven points (0° to 360°) should be used forthe derivatives.

The formulas for the first and second derivatives are as follows:${r^{\prime}(\theta)} = {\sum\limits_{j = 1}^{m}\left( {{{- A_{1}}j\;{\sin({j\theta})}} + {B_{1}j\;{\cos({j\theta})}}} \right)}$${r^{\prime\prime}(\theta)} = {\sum\limits_{j = 1}^{m}\left( {{{- A_{1}}j^{2}{\cos({j\theta})}} - {B_{1}j^{2}{\sin({j\theta})}}} \right)}$

A commercial root-finding subroutine called ZREAL, which is available aspart of the International Mathematical Subroutine Library, from VisualNumerics, Inc., of Houston, Tex., preferably is used to find the rootsof the first derivative—i.e., the points at which the first derivativeis equal to zero. This subroutine requires that initial guesses as tothe number and locations of the roots be provided. Although it is knownthat there are normally eight roots (four maxima and four minima), itwas found that guessing eight roots (which were arbitrarily guessed tobe equiangularly spaced) did not find all roots. It was found insteadthat guessing that there are twenty equiangularly spaced roots yieldsthe correct results. However, it was also found that for higher-orderfits (m≧10), the curve fit so well that variations in the data createdwiggles in the fitted curve that were read as local extrema, yieldingadditional roots. Therefore, m preferably should be less than 10; mostpreferably, m=7.

Once the roots are found, identifying the extrema, the second derivativeat each root is taken, and those points at which the second derivativeis positive are identified as minima. The lines connecting pairs ofoppositely spaced minima are the planar oscillation planes, and the loaddata preferably are used as discussed above to identify the principalplanar oscillation plane.

As stated above, vibration frequency is also recorded at each angularposition. As also set forth above, the stiffness of the shaft can bederived from the vibration frequency using the relationship:f≈(k/M)^(0.5),where k, the spring constant of the shaft in its transverse bendingmode, is a measure of the shaft stiffness. M, the mass, is the totalmass of the oscillating system, which, in this case, is the golf clubshaft 110 plus the tip mass and sensor assembly 1877. Approximating golfclub shaft 110 as a prismatic beam (i.e., a beam of constant crosssection, which most golf club shafts in fact are not) of mass m_(shaft)and assigning the tip mass a mass m_(tip), then the total mass, M, inthe relationship above can be approximated as M=0.23m_(shaft)+m_(tip).Therefore, the frequency can be approximated as:f=(k/(0.23m _(shaft) +m _(tip)))^(0.5).Solving for k yields:k=(0.23m _(shaft) +m _(tip))f ².

Determining k provides a measurement of stiffness by which one shaft canbe compared to another (assuming the same length of shaft is vibratingas discussed above). Determining k also allows one to determine thetip-to-butt deviation of the shaft based on the restoring forcemeasurements gathered during the load test at the various angles. Thisdeviation also can be determined using a travel gauge, or by usingoptical techniques, if preferred.

As set forth above, at each angle:F/k=d+δ,where F is the measured restoring force and d is the displacementapplied during the load test. If k is also known, δ, the deviation ofthe center of the shaft tip from a longitudinal axis passing through thecenter of the shaft butt, can be determined.

In an alternative and more particularly preferred embodiment, thedeviation δ can be determined during a modified load test without firstdetermining stiffness (as measured by the spring constant k). In thisembodiment, cylinder 191 raises shaft 110 a first displacement d₁ andrestoring force data are collected, and then cylinder 191 raises shaft110 to a second displacement d₂ and restoring force data are againcollected. The two restoring force data points for each angular positioncan be captured by moving cylinder 191 up and down at each angularposition at which oscillation data are collected. More preferably, therestoring force data points are captured as part of a modified versionof the load test described above, in which cylinder 191 is moved toposition d₁ and shaft 110 is rotated through at least 360° while dataare captured, with cylinder 191 then being moved to position d₂ andshaft 110 again rotated through at least 360° while again data arecaptured. In a particularly preferred embodiment, data are captured atdisplacement d₁ as shaft 110 is rotated in a first direction through atleast 360°—e.g., through about 400°, and data are again captured atdisplacement d₂ as shaft 110 is rotated in a second, opposite directionthrough the same total angular displacement. This provides two equationsin two unknowns—k and δ—which can be solved for δ:F ₁ =k(d ₁+δ)F ₂ =k(d ₂+δ)k=k,∴F ₁/(d ₁+δ)=F ₂/(d ₂+δ)F ₁ d ₂ +F ₁ δ=F ₂ d ₁ +F ₂δδ=(F ₂ d ₁ −F ₁ d ₂)/(F ₁ −F ₂)

Because the load test is being carried out with the tip mass attached,the weight of the tip mass preferably is subtracted from the measuredrestoring force. The load test data measured at a particular positionare recorded for a different position 90° from the particular positionat which the measurement is being taken, to account for the fact thatthe load test is conducted vertically while the planar oscillation planelocation measurement for the same angular position is carried outhorizontally.

Typically, when δ is plotted as a function of angle, the result can berepresented, as shown in FIG. 22, a larger circle 214 centered on theorigin 215 and a smaller circle 216 offset from origin 215. Therelationship of the diameters of the larger and smaller circles isproportional to the relationship of the diameters of the butt and tipends of shaft 110, and in a preferred embodiment the diameters of therespective circles are equal to the diameters of the respective ends.Thus, this plot represents the location of the tip end relative thelongitudinal axis of shaft 110, or in other words the extent to whichshaft 110 is not straight. The line 217 represents the direction of thebend. That the restoring force data from the load test provides thisresult is to be expected. If a golf club shaft is bent in a particulardirection, then applying a force during a load test in the direction ofthe bend will result in a smaller restoring force than if the appliedforce is applied against the direction of the bend. Thus, for eachangle, if the restoring force is relatively small, then for an angle180° away, the restoring force will be relatively large, and vice-versa.

The process and apparatus according to the present invention can be usedas part of a larger process or apparatus for assembling golf clubs, toproduce “spine-aligned” golf clubs. Thus, each golf club shaft 110,having been marked with a reference mark at a predetermined locationrelative to the location of the preferred orientation, or planaroscillation plane (whether or not marked to indicate the “hard” side),can be passed to a golf club assembly station where the marking on theshaft is identified and used to assemble a golf club with the planaroscillation plane preferably substantially perpendicular to the golfclub face. Depending on the relative speeds of planar oscillation planelocating apparatus 60 or 1870 as compared to the golf club assemblystation, more or fewer planar oscillation plane locating stations orassembly stations, as may be appropriate, can be provided. Thus, severalplanar oscillation plane location stations 60, 1870 may be used to feeda single golf club assembly station. A hopper may be provided at thegolf club assembly station to act as a buffer in case the assemblystation slows down or stops, or is not ready to accept a new golf clubshaft 110 the moment the shaft arrives.

The golf club assembly station preferably is equipped with a scanner foridentifying the mark made on golf club shaft 110 indicating the locationof the planar oscillation plane. Once that mark has been identified,shaft 110 is rotated so that the mark is in a predetermined orientationfor the type of golf club head to be attached to shaft 110, and thatgolf club head is held in a predetermined orientation as shaft 110 isassembled to the golf club head.

Alternatively, each golf club head could be provided with an alignmentmarking to which the marking on golf club shaft 110 must be matched. Ascanner scans for the alignment marks on both shaft 110 and the golfclub head and rotates shaft 110 until the two markings are aligned. Thiseliminates the need for the golf club head holding mechanism to “know” aspecific orientation in which to hold each different type of golf clubhead for alignment with the marked shaft. Instead, each golf club headcan be held in the same orientation, and as shaft 110 is brought closefor assembly, shaft 110 can be rotated until the marking on shaft 110and the marking on the golf club head are in desired alignment beforeshaft 110 is joined to the golf club head.

Apparatus 220 for assembling golf clubs in accordance with the presentinvention is shown in FIGS. 23 and 24. Apparatus 220 includes at leastone apparatus 60 or 1870 (one apparatus 60 shown), a conveyor 221 forremoving completed shafts 110 from apparatus 60, 1870 and depositingthem in a hopper 222, a feed mechanism 223 for feeding each shaft 110from hopper 222 to assembly station 224, and assembly station 224itself.

At assembly station 224, a feeder including arms 225 connected to amotor (not shown) delivers shaft 110 to chuck 230, similar to chucks 76,1876 which rotatably holds shaft 110 from its proximal end. Gripper 231holds a golf club head 232, which may or may not bear an alignmentmarking 233; if there is no alignment marking 233, golf club head 232 isheld by gripper 231 in a known position, which may differ for differenttypes of golf club heads. A scanner 234 scans shaft 110 for marking 235as chuck 230 rotates. When scanner 234 identifies marking 235, processor61 instructs chuck 230 to align marking 235 with alignment marking 233located by scanner 236, or with a predetermined orientation for golfclub head 232. Chuck 230 and gripper 231 are then moved together bymoving one or both thereof, and shaft 110 is joined to golf club head232 in an otherwise conventional way, using whatever adhesives,ferrules, etc. as may be necessary.

FIG. 25 is a sample of a printout that may be provided to a customer inthe case of a golf club retrofit, giving various characteristics of theshaft and comparing the club's original configuration to its newconfiguration. This printout provides information for the consumerregarding the characteristics of the golf club, and also provides a databank of information for the retrofitter regarding each club that hasbeen retrofitted.

Although the data are laid out in a particular arrangement in FIG. 25,other arrangements are possible and would be within the scope of theinvention. Customer and shaft identification data are preferablyprovided in field 250. Included in the identification data preferably isa bar code or other machine-readable indicium (not shown), which may beplaced in box 258 of field 250 and which can be used to recall data forthe particular shaft from a data repository. A matching bar code orother indicium could be applied to the shaft itself. In particular, if alabel is used to apply alignment markings to the shaft as describedabove, that label could also bear the indicium.

The printout preferably includes a graph 251 showing the results of theload test discussed above. In particular, the load symmetry index (LSI)discussed above is reported, and the normalized load during the loadtest is correlated to stiffness in foot-pounds per inch. The results ofthe “spining” or planar oscillation plane location measurements areshown at 252. In particular, two phase plots 253, 254 are presented toshow, respectively, the shaft vibration characteristics in the “logo-up”position and in the principal planar oscillation plane as located. Aplot 255, similar to FIG. 21, is also provided, except that while lines259, 260 representing all planar oscillation planes preferably aredisplayed, preferably line 259 representing the principal planaroscillation plane is heavier or otherwise differentiated from any otherline 260. Similarly, a plot 256, like that of FIG. 22, is also providedto show the straightness of the shaft, and plot 257 showing vibrationfrequency (a measure of stiffness) as a function of angular position isalso provided. In plot 257, circular data points represent a “perfect”shaft in which the stiffness, and hence the frequency, is the same atall angles, while the square data points show the frequency data for theshaft being measured.

While the invention has been described so far in terms of golf clubshafts, it can be used to determine the symmetry/asymmetry, roundness,straightness and/or stiffness of any elongated member, including, butnot limited to, baseball bats, billiard cues, arrows, fishing rods, orany structural member.

Thus it is seen that a method and apparatus for quickly and reliablydetermining the preferred angular orientation of a golf club shaft orother elongated member, and for using the determination of the preferredangular orientation to automatically assemble golf clubs with eachrespective golf club shaft consistently aligned relative to therespective club face, are provided. One skilled in the art willappreciate that the present invention can be practiced by other than thedescribed embodiments, which are presented for purposes of illustrationand not of limitation, and the present invention is limited only by theclaims which follow.

What is claimed is:
 1. A method of determining straightness of a golfclub shaft, said shaft having a handle end and having a tip end formating to a club head, said method comprising: immobilizing said handleend of said golf club shaft and defining a longitudinal axis passingthrough said handle end and extending perpendicularly with respect to aplane perpendicular to said handle end; determining a spring constant ofsaid golf club shaft in a transverse bending mode; and for each of aplurality of angles about said longitudinal axis: displacing said tipend of said shaft transversely to, and by a predetermined distance from,said longitudinal axis, measuring a restoring force during saiddisplacing, determining a difference between said measured restoringforce during said displacing and an expected restoring force based onsaid predetermined distance and said spring constant, and deriving fromsaid difference and said spring constant a deviation of said tip endfrom said longitudinal axis at said angle.
 2. The method of claim 1further comprising plotting said deviations at said plurality of angles,thereby providing a visual representation of said straightness of saidshaft.
 3. The method of claim 2 wherein said determining said springconstant comprises: initiating vibration of said shaft in saidtransverse bending mode; measuring frequency of said vibration; andderiving said spring constant from said frequency.
 4. The method ofclaim 1 wherein said determining said spring constant comprises:initiating vibration of said shaft in said transverse bending mode;measuring frequency of said vibration; and deriving said spring constantfrom said frequency.
 5. A method of determining straightness of a golfclub shaft, said shaft having a handle end and having a tip end formating to a club head, said method comprising: immobilizing said handleend of said golf club shaft and defining a longitudinal axis passingthrough said handle end and extending perpendicularly with respect to aplane perpendicular to said handle end; and for each of a plurality ofangles about said longitudinal axis: initially displacing said tip endof said shaft transversely to, and by a first predetermined distancefrom, said longitudinal axis, measuring a first restoring force duringsaid initial displacing, subsequently displacing said tip end of saidshaft transversely to, and by a second predetermined distance from, saidlongitudinal axis, measuring a second restoring force during saidsubsequent displacing, and deriving, from said first and secondrestoring forces and said first and second predetermined distances, adeviation of said tip end from said longitudinal axis at said angle. 6.The method of claim 5 further comprising plotting said deviations atsaid plurality of angles, thereby providing a visual representation ofsaid straightness of said shaft.
 7. Apparatus for determiningstraightness of a golf club shaft, said shaft having a handle end andhaving a tip end for mating to a club head, said apparatus comprising:means for immobilizing said handle end of said golf club shaft anddefining a longitudinal axis passing through said handle end andextending perpendicularly with respect to a plane perpendicular to saidhandle end; means for determining a spring constant of said golf clubshaft in a transverse bending mode; and means for, for each of aplurality of angles about said longitudinal axis: displacing said tipend of said shaft transversely to, and by a predetermined distance from,said longitudinal axis, measuring a restoring force during saiddisplacing, determining a difference between said measured restoringforce during said displacing and an expected restoring force based onsaid predetermined distance and said spring constant, and deriving fromsaid difference and said spring constant a deviation of said tip endfrom said longitudinal axis at said angle.
 8. The apparatus of claim 7further comprising means for plotting said deviations at said pluralityof angles, thereby providing a visual representation of saidstraightness of said shaft.
 9. The apparatus of claim 8 wherein saidmeans for determining said spring constant comprises: means forinitiating vibration of said shaft in said transverse bending mode;means for measuring frequency of said vibration; and means for derivingsaid spring constant from said frequency.
 10. The apparatus of claim 7wherein said means for determining said spring constant comprises: meansfor initiating vibration of said shaft in said transverse bending mode;means for measuring frequency of said vibration; and means for derivingsaid spring constant from said frequency.
 11. Apparatus for determiningstraightness of a golf club shaft, said shaft having a handle end andhaving a tip end for mating to a club head, said apparatus comprising:means for immobilizing said handle end of said golf club shaft anddefining a longitudinal axis passing through said handle end andextending perpendicularly with respect to a plane perpendicular to saidhandle end; and means for, for each of a plurality of angles about saidlongitudinal axis: initially displacing said tip end of said shafttransversely to, and by a first predetermined distance from, saidlongitudinal axis, measuring a first restoring force during said initialdisplacing, subsequently displacing said tip end of said shafttransversely to, and by a second predetermined distance from, saidlongitudinal axis, measuring a second restoring force during saidsubsequent displacing, and deriving, from said first and secondrestoring forces and said first and second predetermined distances, adeviation of said tip end from said longitudinal axis at said angle. 12.The apparatus of claim 11 further comprising means for plotting saiddeviations at said plurality of angles, thereby providing a visualrepresentation of said straightness of said shaft.
 13. Apparatus fordetermining straightness of a golf club shaft, said shaft having ahandle end and having a tip end for mating to a club head, saidapparatus comprising: a clamp for immobilizing said handle end of saidgolf club shaft and defining a longitudinal axis passing through saidhandle end and extending perpendicularly with respect to a planeperpendicular to said handle end; an analyzer to determine a springconstant of said golf club shaft in a transverse bending mode; and adeviation calculator for, for each of a plurality of angles about saidlongitudinal axis: displacing said tip end of said shaft transverselyto, and by a predetermined distance from, said longitudinal axis,measuring a restoring force during said displacing, determining adifference between said measured restoring force during said displacingand an expected restoring force based on said predetermined distance andsaid spring constant, and deriving from said difference and said springconstant a deviation of said tip end from said longitudinal axis at saidangle.
 14. The apparatus of claim 13 further comprising a plotter toplot said deviations at said plurality of angles, thereby providing avisual representation of said straightness of said shaft.
 15. Theapparatus of claim 14 wherein said spring constant analyzer comprises: avibration generator for initiating vibration of said shaft in saidtransverse bending mode; a frequency counter for measuring frequency ofsaid vibration; and a processor for deriving said spring constant fromsaid frequency.
 16. The apparatus of claim 13 wherein said springconstant analyzer comprises: a vibration generator for initiatingvibration of said shaft in said transverse bending mode; a frequencycounter for measuring frequency of said vibration; and a processor forderiving said spring constant from said frequency.
 17. Apparatus fordetermining straightness of a golf club shaft, said shaft having ahandle end and having a tip end for mating to a club head, saidapparatus comprising: a clamp for immobilizing said handle end of saidgolf club shaft and defining a longitudinal axis passing through saidhandle end and extending perpendicularly with respect to a planeperpendicular to said handle end; and a deviation calculator for, foreach of a plurality of angles about said longitudinal axis: initiallydisplacing said tip end of said shaft transversely to, and by a firstpredetermined distance from, said longitudinal axis, measuring a firstrestoring force during said initial displacing, subsequently displacingsaid tip end of said shaft transversely to, and by a secondpredetermined distance from, said longitudinal axis, measuring a secondrestoring force during said subsequent displacing, and deriving, fromsaid first and second restoring forces and said first and secondpredetermined distances, a deviation of said tip end from saidlongitudinal axis at said angle.
 18. The apparatus of claim 17 furthercomprising a plotter for plotting said deviations at said plurality ofangles, thereby providing a visual representation of said straightnessof said shaft.